Donovan s Conjecture for certain defect groups

Donovan’s Conjecture for certain defect groups

(joint work with C. W. Eaton and B. Külshammer)

Donovan’s Conjecture

Let D be a finite p-group for a prime number p.

Donovan’s Conjecture

There are only finitely many Morita equivalence classes of blocks of finite groups with defect group D.

A weak version is the following.

Conjecture

There is a bound on the Cartan invariants of blocks of finite groups with defect group D which only depends on D.

Known results

If D is a cyclic group, then Donovan’s Conjecture is true. If p = 2 and D has maximal class, all blocks with defect group D have tame representation type.

Then, by Erdmann’s work the Morita equivalence class of a block with defect group D is known up to certain parameters. Donovan’s Conjecture also holds if one restricts to blocks of p-solvable, symmetric or alternating groups.

Hiss and Kessar showed Donovan’s Conjecture for some blocks of classical groups.

Rédei’s classification

In the following we consider blocks with respect to a splitting p-modular system (K, O, F).

Assume that p = 2 and D is a minimal nonabelian 2-group.

This means all proper subgroups of D are abelian, but D is not. Then by a result of Rédei D is isomorphic to one of the following groups:

(a) hx, y | x2r

= y2s = 1, xyx−1 = y1+2s−1i with r ≥ 1 and s ≥ 2,

(b) hx, y | x2r

= y2s = [x , y ]2 = [x , x , y ] = [y , x , y ] = 1i with 2 ≤ r ≥ s ≥ 1,

(c) Q8.

Rédei’s classification

Let B be a block of a finite group with defect group D as given above.

If D is metacyclic, then B is nilpotent unless D ∼= D8or D ∼= Q8. In the nilpotent case B is Morita equivalent to the group algebra OD by Puig’s Theorem.

For D ∼= D8 or D ∼= Q8 we can apply Erdmann’s work.

Hence, we may assume that case (b) in Rédei’s classification occurs, i. e.

D := hx, y | x2r = y2s = [x , y ]2= [x , x , y ] = [y , x , y ] = 1i,

where 2 ≤ r ≥ s ≥ 1, [x, y ] := xyx−1y−1 and [x, x, y ] :=

The case s = 1

Assume that B is not nilpotent. Then it turns out that s = 1 or r = s.

Let ki(B) be the number of ordinary irreducible characters of

height i ≥ 0 of B, and let k(B) =P∞

i =0ki(B).

Similarly l (B) is the number of irreducible Brauer characters of B.

For s = 1 these block invariants are given by the following theorem.

The case s = 1

Theorem (S., 2010)

Let B be a non-nilpotent block of a finite group with defect group D = hx, y | x2r = y2= [x , y ]2= [x , x , y ] = [y , x , y ] = 1i for some r ≥ 2. Then

k(B) = 5 · 2r −1, k0(B) = 2r +1, k1(B) = 2r −1, l (B) = 2 and the Cartan matrix of B is given by

2r −13 1

1 3 

D minimal nonabelian

Rédei (a) Rédei (b) D ∼= Q8

D ∼= D8 s = 1 nilpotent block r = s Donovan’s Conjecture block invariants EKS Brauer metacyclic Erdmann Olsson Erdmann S. Puig

The case r = s

In the r = s case we were able to prove Donovan’s Conjecture:

Theorem

Let B be a non-nilpotent block of a finite group with defect group D = hx, y | x2r = y2r = [x , y ]2 = [x , x , y ] = [y , x , y ] = 1i for some r ≥ 2. Then B is Morita equivalent to O[D o E ] where E is a subgroup of Aut(D) of order 3. In particular, we have

k(B) = 5 · 2 2r −2_{+ 16} 3 , k0(B) = 22r+ 8 3 , k1(B) = 22r −2+ 8 3 and l (B) = 3.

Sketch of the proof (1)

It turns out that the claim holds for solvable groups. Now the idea is to reduce the situation to quasisimple groups.

Lemma

Let G be a finite group with Sylow 2-subgroup D as given above. Then G is solvable.

Proof.

By Feit-Thompson we may assume O20(G ) = 1.

Then the Z∗-Theorem implies z := [x, y ] ∈ Z(G ). Thus, G /hzi has Sylow 2-subgroup D/hzi ∼= C2r × C₂r.

Sketch of the proof (2)

Let G be a finite group with a non-nilpotent block B as above. Then by Fong reduction we may assume that O20(G ) is cyclic

and central.

An application of the Külshammer-Puig Theorem gives O2(G ) ⊆ D0= hzi

and Z(G ) = F(G ).

It turns out that B covers a non-nilpotent block b of the layer E(G ) of G with defect group D.

Moreover, b covers a non-nilpotent block of a component of G also with defect group D.

Sketch of the proof (3)

Hence, by way of contradiction we may assume that G is quasi-simple, i. e. G0= G and G / Z(G ) is simple.

Now we apply the classification of the finite simple groups. By the lemma above, 64 (≤ 2|D|) divides |G |.

If G / Z(G ) is an alternating group, the situation is very easy, since one can use the representation theory of symmetric groups. For the non-principal 2-blocks of the (covering groups of the) sporadic groups results of Landrock and An-Eaton can be used.

Sketch of the proof (4)

Thus, it remains to deal with the simple groups of Lie type. By a result of Humphreys it suffices to consider Lie groups in odd characteristic.

Here one can use methods going back to Deligne, Lusztig and others.

We illustrate these for the case G / Z(G ) ∼= PSL(n, q).

Here one can go over to H := GL(n, q) (ignoring exceptional covers).

Sketch of the proof (5)

Then there is a semisimple element s ∈ H such that a Sylow 2-subgroup of C_H(s) is related to D.

In particular it can be shown that CH(s) is solvable.

On the other hand CH(s) has the form

CH(s) ∼= t

×

i =1

GL(ni, qmi).

Proposition

In the situation of the last theorem, all simple B-modules have vertex D.

Proof.

The result holds for G = D o E , since the irreducible Brauer characters are restrictions of ordinary characters of height 0 in this case.

Since Morita equivalence preserves decomposition matrices, the result follows for arbitrary G .

Corollary

For a 2-block B of a finite group with minimal nonabelian defect group the following conjectures are satisfied:

Alperin’s Weight Conjecture Brauer’s k(B)-Conjecture Brauer’s Height-Zero Conjecture Dade’s Ordinary Conjecture Alperin-McKay Conjecture Olsson’s Conjecture Eaton’s Conjecture Eaton-Moretó Conjecture Malle-Navarro Conjecture

Background

For primes p > 3, Héthelyi, Külshammer and myself proved Olsson’s Conjecture for p-blocks with defect groups of p-rank 2.

In the process the extraspecial defect group D of order 53 and exponent 5 turns out to be very difficult.

In particular blocks with defect group D and a specific fusion system are complicated.

We end up by determining the Morita equivalence class of such blocks.

A result

Proposition

Let B be a block of a finite group G with an extraspecial defect group

D of order 53 and exponent 5. Suppose that the fusion system of B

is the same as the fusion system of the sporadic simple Thompson group Th for the prime 5. Then B is Morita equivalent to the principal 5-block of Th.

Sketch of the proof (1)

We observe that all nontrivial elements of D are conjugate in the fusion system.

By Fong reduction we may assume F(G ) = Z(G ) = O50(G ).

It turns out that G has only one component, i. e. E(G ) is quasi-simple.

So S := E(G )/ Z(E(G )) is simple and

G / Z(G ) ≤ Aut(E(G )) ≤ Aut(S). B covers a block b of E(G ) with defect group D.

Sketch of the proof (2)

For the block b we use An and Eaton’s classification of blocks of quasisimple groups with extraspecial defect groups.

This shows that D must be a Sylow 5-subgroup of E(G ). But then in most cases there are elements x, y ∈ S of order 5 such that |CS(x )| 6= |CS(y )|.

Sketch of the proof (3)

In particular, x and y cannot be conjugate in G . This contradicts the structure of the fusion system.

The only remaining case is S = Th and b = B0(E(G )).

Fortunately here we have Out(Th) = M(Th) = 1, so that G = S × Z(G ).